## 3.4. Lim Sup and Lim Inf

### Examples 3.4.2(b):

Since this sequence is*{1, 1/2, 1/3, 1/4, ...}*the infimum is zero, while the supremum is 1. As for

*lim inf*and

*lim sup*, we find first the sequence of numbers

*A*and

_{j}*B*mentioned in the definition.

_{j}*A*_{1}= inf{1, 1/2, 1/3, 1/4, ...} = 0*A*_{2}= inf{1/2, 1/3, 1/4, 1/5, ...} = 0*A*_{3}= inf(1/3, 1/4, 1/5, 1/6, ...} = 0

Similarly, we find the numberslim inf = 0

*B*

_{j}= sup{a_{j}, a_{j + 1}, a_{j + 2}, ...}:*B*_{1}= sup{1, 1/2, 1/3, 1/4, ...} = 1*B*_{2}= sup{1/2, 1/3, 1/4, 1/5, ...} = 1/2*B*_{3}= sup(1/3, 1/4, 1/5, 1/6, ...} = 1/3

lim sup = 0